Summer workbook — GCSE Higher

1.Work out 35% of 180.

2.Work out the exact value of √(2² + 3²)

3.The decimal 0.2333... has one non-recurring digit (the 2) followed by a single recurring digit (the 3), so it can be written as 0.2 recurring 3. Let x = 0.2333... . Work out x as a fraction in its simplest form.

4.Ten athletes run in a final. Gold, silver and bronze medals are awarded to the first three athletes to finish, and there are no ties. Work out how many different ways the three medals can be awarded.

5.A length, L cm, has the error interval 24.5 ≤ L < 25.5. Write down the degree of accuracy to which the length was measured.

6.Work out 5/6 × 2/9, giving your answer in its simplest form.

7.A carton of orange juice holds 1.35 litres. Ruby pours the juice equally into 4 identical glasses. Work out how much juice is in each glass, giving your answer as a fraction of a litre in its simplest form.

8.Without using a calculator, estimate the value of √70 × ∛65, giving your answer to 1 significant figure.

9.Four locations each record a temperature one winter morning. Which of these temperatures is the coldest?

10.Work out √3 × √12, giving your answer as an integer.

1.Solve the simultaneous equations y = 2x − 1 and y = x² − x − 1, giving both pairs of solutions.

2.A straight line has equation 4y = 12x + 20. Work out the gradient of the line.

3.Solve the simultaneous equations 2x + y = 7 and x − y = 2.

4.Yusuf has £40 saved and adds £6 each week. His sister Aisha has £10 saved and adds £11 each week. After how many weeks will they have saved the same amount?

5.Solve (3x + 2)/5 = (x + 4)/3

6.Solve the inequality 9 − 2x ≥ 1.

7.Work out the gradient of the straight line with equation 2x + y = 8.

8.The nth term of a sequence is n² + 2n − 4. Work out the 7th term.

9.Two numbers have a sum of 50. The larger number is twice the smaller number. Work out the smaller number.

10.A rule turns each input into an output. An input of 0 gives an output of −1, an input of 1 gives an output of 1, and an input of 2 gives an output of 3. Work out the rule, writing the input as x and the output as y.

1.A mobile phone tariff is shown on a straight-line graph with the monthly cost, C pounds, on the vertical axis and the amount of data used, g gigabytes, on the horizontal axis. The line passes through (0, 10) and (8, 26). Work out the gradient and say what it represents.

2.Write £3.60 : £2.40 as a ratio in its simplest form.

3.The volume of fuel remaining in a storage tank, in thousands of litres, is plotted against time, in hours, since a leak was detected. A tangent to the graph at t = 2 hours has gradient −8.75 thousand litres per hour. A tangent at t = 8 hours has gradient −3.5 thousand litres per hour. Work out by what factor the instantaneous rate of change is greater in size at t = 2 hours than at t = 8 hours.

4.A baker uses 450 g of flour and 300 g of butter in a batch of pastry. Write the mass of butter as a fraction of the total mass of flour and butter, in its simplest form.

5.In a fruit drink, cranberry juice and apple juice are mixed in the ratio 3:7. Write the amount of apple juice as a fraction of the amount of cranberry juice, in its simplest form.

6.Triangle ABC is mathematically similar to triangle PQR, with AB corresponding to PQ and BC corresponding to QR. AB = 6 cm, BC = 8 cm and PQ = 12 cm. Work out the length of QR.

7.A statue exerts a downward force of 68.4 N on its base, which has an area of 2.85 m². Work out the pressure that the statue exerts on its base, in N/m².

8.A vehicle is worth £18000. Its value decreases by 15% each year. Using the recurrence V_{n+1} = 0.85V_n, with V_0 = 18000, find the first whole number of years after which the vehicle's value drops below £10000.

9.The density of a liquid is 0.92 g/cm³. Work out the density of the liquid in kg/m³.

10.Two mathematically similar rectangles have lengths in the ratio 3 : 5. The area of the smaller rectangle is 27 cm². Work out the area of the larger rectangle.